King Fahd University of Petroleum & Minerals. Electrical Engineering Department. EE 575 Information Theory

Transcription

1 King Fahd University of Petroleum & Minerals Electrical Engineering Department EE 575 Information Theory BER Performance of VBLAST Detection Schemes over MIMO Channels Ali Abdullah Al-Saihati ID# Sec# 1 Abstract Theory V-BLAST algorithm and the different types of it will be discussed in this report. Three types of detection schemes will be discussed here as well: MMSE and ZF and ML detections. Also, the problem of the project will be defined and explained. MatLab code for the simulation will be included and the results will be shown. Finally, the observations for the simulation results will be placed in the conclusion. Due on 8/6/2010

2 TABLE OF CONTENTS I. Introduction... 1 II. V-BLAST Theory... 1 III. Problem Definition...10 IV. MatLab Simulation a) MatLab Code b) Results V. Conclusion...14 VI. References...15 II

3 I. Introduction: MIMO system has proved to achieve high capacity compared to SISO MISO and SIMO systems. For this reason, many algorithms have been proposed to reduce the interference in the received signals caused by other transmitters in the system. Also, they aim achieve closer values to the Shannon capacity limit. D-BLAST (Diagonal Bell Labs Layered Space Time) and V-BLAST (Vertical Bell Labs Layered Space Time) are such schemes used for detection and suppression the interference in MIMO systems. D-Blast, which was proposed by Gerard J. Foschini, applies a diagonal space time coding on the data. By applying this algorithm, it could achieve 90% of Shannon capacity rates as well as high spectral efficiency. However, due to complexity of implementing the algorithm, V-Blast algorithm was proposed. It was established in 1996 at Bell Labs. It demultiplexes the transmitted signal and then maps bit to symbol independently for each substream. II. V-Blast Theory: V-Blast is a single user scheme which has multiple transmitters. It divides the data stream into substreams and transmits them through multiple transmitters at the same time and frequency. This results in receiving the data at the receiver at the same time and frequency. By implementing V- BLAST algorithm, the diversity gain is increased and the bit error rate (BER) performance is improved. The MIMO system is assumed to undergoes flat fading channel. The system model of the output signal is given by: y= Hx+ η Where y is the received signal, x is the transmitted signal, η is the added noise and H is the channel model of the system. Figure 1 shows V-BLAST system. 1

4 Figure 1: V-BLAST system [2] The detection process consists of three operations: interference suppression (nulling), interference cancellation (subtraction) and optimal ordering. The interference nulling process is carried out by projecting the received signal into the null subspace spanned by the interfering signals. This process is done by using Gramm-Schmidt orthogonalization procedure that converts a set of linearly independent vectors into orthogonal set of vectors. Then, the symbol is detected. The interference cancellation process is done by subtracting the detected symbol from the received signal. The optimal ordering, which is last process, ensures that the detected symbol has the highest signal to noise ratio (SNR). So, V-BLAST algorithm integrates both, linear and nonlinear algorithms presented in interference nulling and interference cancellation respectively. Figure 2 shows the architecture of the successive interference cancellation while Figure 3 shows the perpendicular and parallel components of the received signal with subspace. However, there are two disadvantages in V-BLAST algorithms. The first one is that the error propagates during symbol detection. The other one is that the number of receive antennas must be greater than or equal to the number of transmit antennas to satisfy the interference nulling process. 2

5 Figure 2: architecture of the successive interference cancellation. [5] Figure 3: perpendicular and parallel components of the received signal with the subspace. [3] 3

6 The detection algorithm is given by: Where G, w are the weight vector, r is the received vector before estimation, H + is the pseudo inverse of the channel matrix, q is the decision process and i is the increment index. Since the amount of interference cancelled in each step becomes smaller, a new algorithm was proposed which is called new V- BLAST algorithm. Here, the algorithm stops iterating when the interference becomes very small. Hence, it reduces complexity of the system. Figure 4 shows the flow chart of a modified V-BLAST algorithm where C is a constant. When the value of C becomes 1, the algorithm becomes the same as the original V-BLAST detection. When C becomes 0, on the other hand, the algorithm becomes minimum mean square error (MMSE) and zero forcing (ZF) detection. 4

7 Figure 4: flow chart of a modified V-BLAST algorithm [5] The weight vector for the ZF and MMSE are given by: G ZF =H + = ( H H H ) -1 H H G MMSE =H + = ( H H H + ρ I ) -1 H H Where I is the identity matrix and ρ is the SNR. While ZF and MMSE are simple to implement because they are linear algorithms, they do achieve high data rate at high SNR. The ZF detection cancels the interference only So, it enhances the noise in each iteration. At high SNR, the MMSE detection will function like ZF detection. The maximum likelihood (ML) detection is given by: 5

8 G = min y H x The ML is optimum in minimizing the error. Although it has an excellent performance, the complexity is high. The order of complexity is A M where M is the number of transmitter and A is the number of modulation constellation. For example, if M = 10 and A = 2 then we need to compute 1024 times during the process. Different proposed recursive algorithms have been proposed for V-BLAST algorithm. Some of these are matrix recursive, vector recursive, greedy ordering, scalar recursive and adaptive scalar recursion for fast fading. The matrix recursive algorithm tries to find an inverse matrix using the Sherman- Morrison formula with a given initial matrix recursively. This method decreases the complexity order from quadric to cubic but the computation of the inverse matrix is complex. In vector recursive algorithm, a weight vector is found recursively to substitute the computation of inverse matrix. The greedy ordering method selects the most reliable signals for detection. The scalar recursion algorithm focuses on nulling the output vector. The adaptive scalar recursion for fast fading changes and updates the weight vectors and optimum ordering based on the changes incurred during transmission. Using this algorithm, the complexity order reduces to a square. Figure 5 shows the asymptotic complexity in multiplications as a function of number of antenna pairs and Table 1 shows total number of real valued arithmetic operations required for V-BLAST algorithms. Figure 6 and Figure 7 shows the BER performance of the different proposed algorithms using 4QPSK and 16QAM respectively. Table 1 shows total number of real valued arithmetic operations required for V-BLAST algorithms. [5] 6

9 Figure 5: the asymptotic complexity in multiplications as a function of number of antenna pairs [5] 7

10 Figure 6: BER performance of the different proposed algorithms using 4QPSK [5] 8

11 Figure 7: BER performance of the different proposed algorithms using 16QAM. [5] 9

12 III. Problem Definition: It is required to find the BER performance of the ZF, MMSE and (ML) schemes implemented in the V-BLAST system IV. MatLab Simulation: In this simulation, a 2 x 2 MIMO system with BPSK modulation is used. Then, the algorithm of V-BLAST is implemented. After that, weight vector is produced using ZF, MMSE and ML detectors. Finally, the BER performance is plotted. A flat fading Rayleigh channel is assumed in this simulation. a) MatLab Code clear N = 10^6; % Block size snr = [0:45]; % SNR values MT = 2; % Number of transmitters MR = 2; % Number of Receivers for i = 1:length(snr) p = rand(1,n)>0.5; % Generates random numbers s = 2*p-1; % Converts 0 & 1 t0-1 and 1 smod = kron(s,ones(mr,1)); % smod = reshape(smod,[mr,mt,n/mt]); h = 1/sqrt(2)*[randn(MR,MT,N/MT) + j*randn(mr,mt,n/mt)]; % Rayleigh channel n = 1/sqrt(2)*[randn(MR,N/MT) + j*randn(mr,n/mt)]; % White gaussian noise with 0dB variance y = squeeze(sum(h.*smod,2)) + 10^(-snr(i)/20)*n; % Noise addition % forms weight vector for MMSE hcof = zeros(2,2,n/mt) ; hcof(1,1,:) = sum(h(:,2,:).*conj(h(:,2,:)),1) + 10^(-snr(i)/10); hcof(2,2,:) = sum(h(:,1,:).*conj(h(:,1,:)),1) + 10^(-snr(i)/10); hcof(2,1,:) = -sum(h(:,2,:).*conj(h(:,1,:)),1); % c term hcof(1,2,:) = -sum(h(:,1,:).*conj(h(:,2,:)),1); % b term hden = ((hcof(1,1,:).*hcof(2,2,:)) - (hcof(1,2,:).*hcof(2,1,:))); hden = reshape(kron(reshape(hden,1,n/mt),ones(2,2)),2,2,n/mt); hinv = hcof./hden; hmod = reshape(conj(h),mr,n); ymod = kron(y,ones(1,2)); ymod = sum(hmod.*ymod,1); ymod = kron(reshape(ymod,2,n/mt),ones(1,2)); yhat = sum(reshape(hinv,2,n).*ymod,1); phat = real(yhat)>0; % Hard decision decoding 10

13 end nerr(i) = size(find([p- phat]),2); % Counts the # errors ber1 = nerr/n; % simulated ber for i = 1:length(snr) p = rand(1,n)>0.5; % generating 0,1 with equal probability s = 2*p-1; % BPSK modulation 0 -> -1; 1 -> 0 smod = kron(s,ones(mr,1)); % smod = reshape(smod,[mr,mt,n/mt]); % grouping in [MR,MT,N/MT ] matrix h = 1/sqrt(2)*[randn(MR,MT,N/MT) + j*randn(mr,mt,n/mt)]; % Rayleigh channel n = 1/sqrt(2)*[randn(MR,N/MT) + j*randn(mr,n/mt)]; % white gaussian noise, 0dB variance y = squeeze(sum(h.*smod,2)) + 10^(-snr(i)/20)*n; hcof = zeros(2,2,n/mt) ; hcof(1,1,:) = sum(h(:,2,:).*conj(h(:,2,:)),1); hcof(2,2,:) = sum(h(:,1,:).*conj(h(:,1,:)),1); hcof(2,1,:) = -sum(h(:,2,:).*conj(h(:,1,:)),1); hcof(1,2,:) = -sum(h(:,1,:).*conj(h(:,2,:)),1); hden = ((hcof(1,1,:).*hcof(2,2,:)) - (hcof(1,2,:).*hcof(2,1,:))); hden = reshape(kron(reshape(hden,1,n/mt),ones(2,2)),2,2,n/mt); hinv = hcof./hden; hmod = reshape(conj(h),mr,n); ymod = kron(y,ones(1,2)); ymod = sum(hmod.*ymod,1); ymod = kron(reshape(ymod,2,n/mt),ones(1,2)); yhat = sum(reshape(hinv,2,n).*ymod,1); phat = real(yhat)>0; nerr(i) = size(find([p- phat]),2); end ber2 = nerr/n; % simulated ber for i = 1:length(snr) p = rand(1,n)>0.5; s = 2*p-1; smod = kron(s,ones(mr,1)); smod = reshape(smod,[mr,mt,n/mt]); h = 1/sqrt(2)*[randn(MR,MT,N/MT) + j*randn(mr,mt,n/mt)]; % Rayleigh channel n = 1/sqrt(2)*[randn(MR,N/MT) + j*randn(mr,n/mt)]; % white gaussian noise, 0dB variance y = squeeze(sum(h.*smod,2)) + 10^(-snr(i)/20)*n; % Maximum Likelihood Receiver % if [s1 s2 ] = [+1,+1 ] shat1 = [1 1]; shat1 = repmat(shat1,[1,n/2]); shat1mod = kron(shat1,ones(mr,1)); shat1mod = reshape(shat1mod,[mr,mt,n/mt]); zhat1 = squeeze(sum(h.*shat1mod,2)) ; J11 = sum(abs(y - zhat1),1); % if [s1 s2 ] = [+1,-1 ] shat2 = [1-1]; shat2 = repmat(shat2,[1,n/2]); 11

14 shat2mod = kron(shat2,ones(mr,1)); shat2mod = reshape(shat2mod,[mr,mt,n/mt]); zhat2 = squeeze(sum(h.*shat2mod,2)) ; J10 = sum(abs(y - zhat2),1); % if [s1 s2 ] = [-1,+1 ] shat3 = [-1 1]; shat3 = repmat(shat3,[1,n/2]); shat3mod = kron(shat3,ones(mr,1)); shat3mod = reshape(shat3mod,[mr,mt,n/mt]); zhat3 = squeeze(sum(h.*shat3mod,2)) ; J01 = sum(abs(y - zhat3),1); % if [s1 s2 ] = [-1,-1 ] shat4 = [-1-1]; shat4 = repmat(shat4,[1,n/2]); shat4mod = kron(shat4,ones(mr,1)); shat4mod = reshape(shat4mod,[mr,mt,n/mt]); zhat4 = squeeze(sum(h.*shat4mod,2)) ; J00 = sum(abs(y - zhat4),1); rvec = [J11;J10;J01;J00]; % Finds the minimum from the four possible combinations [u dd] = min(rvec,[],1); ref = [1 1; 1 0; 0 1; 0 0 ]; % Bits mapping phat = zeros(1,n); phat(1:2:end) = ref(dd,1); phat(2:2:end) = ref(dd,2); nerr(i) = size(find([p- phat]),2); end ber3 = nerr/n; % simulated ber semilogy(snr,ber1,'r-',snr,ber2,'b-',snr,ber3,'g-'); % plots the ber for legend('mmse','zf', 'ML'); xlabel('snr (db)'); ylabel('ber'); title('ber Performance of Different V-BLAST Detection Schemes over 2x2 MIMO Channels with Rayleigh Channel'); axis([ ^-5 0.5]) grid on 12

15 b) Results: At a BER of 10-4, the SNRs of ML, MMSE and ZF are 16 db, 31 db and 33 db respectively. We see a huge improvement in using ML detection over MMSE and ZF detections by 15 db. The performance of MMSE detection is better than ZF detection by 2-3 db. 13

16 V. Conclusion: From the above results, it has been observed that the ML detection has better BER performance than the MMSE and ZF detections by 15dB. In Addition, the performance of MMSE detection is better than ZF detection by 2-3 db. Finally, by using the adaptive scalar recursion for fast fading, the complexity order reduces to square and the computation becomes less compared to other techniques. 14

17 VI. References: [1] Nirmalend. B and Rabindranath B. Capacity and V-BLAST Techniques for MIMO Wireless Channel. Journal of Theoretical and Applied Information Technology, [2] P. W. Wolniansky. G. J. Foschini. G. D. Golden and R. A. Valenzuela V-BLAST: An Architecture for Realizing Very High Data Rates Over the Rich-Scattering Wireless Channel. Bell Laboratories. [3] S. Loyka and F. Gagnon. Performance Analysis of the V- BLASTAlgorithm: An Analytical Approach International Zurich Seminar on Wireless Broadband. [4] Taekyu Kim and Sin-Chong Park. Reduced Complexity Detection for V-BLAST D Systems from Iteration Canceling [5] Toshiaki. K. Low-Complexity Systolic V-BLAST Architecture IEEE Transactions on Wireless Communications,